Using the compositor to adjust an image that is displayed in the viewport on a mesh
There is no existing internal solution for this.

However, there are still a few ways to try:

Plan A:

If you just want to adjust something like RGB Curve or Brightness/Satuation/Contrast and so on, you may consider using non-compositing adjustment directly from the Texture tab:

 

Plan B:

I think the closest way to serve such need is:


Use File Output node;
F12 to render(to output composite result);
Load that output image into UV/Image Editor, have it displayed on object (just for 3D View);
From now on, each time you want to update the composite result to 3D view, hit F12, after finishing render, hovering mouse cursor within UV/Image Editor, and press AltR to reload it (or access via Image > Reload Image).


Now, you can then see it displayed in solid/textured solid view, etc. I think this screenshot may demostrate all you need:



Tip: Delete the Render Layer node will ignore scene rendering, which will speed up the file output.

I'm sure the workflow above can be made to a single button or shortcut by scripting a bit. You just need someone that can help in trying this in coding way. (not me, yet :P) All in all, I believe anyone who knows scripts can provide a one-stop solution.